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// $Id: GLaguer.cpp 962 2006-11-07 15:13:34Z privmane $
#include "definitions.h"
#include "GLaguer.h"
#include "errorMsg.h"
#include "gammaUtilities.h"
GLaguer::GLaguer(const int pointsNum, const MDOUBLE alf, Vdouble & points, Vdouble & weights)
{
gaulag(_points, _weights, alf, pointsNum);
weights = _weights;
points = _points;
}
//Input: alf = the alpha parameter of the Laguerre polynomials
// pointsNum = the polynom order
//Output: the abscissas and weights are stored in the vecotrs x and w, respectively.
//Discreption: given alf, the alpha parameter of the Laguerre polynomials, the function returns the abscissas and weights
// of the n-point Guass-Laguerre quadrature formula.
// The smallest abscissa is stored in x[0], the largest in x[pointsNum - 1].
void GLaguer::gaulag(Vdouble &x, Vdouble &w, const MDOUBLE alf, const int pointsNum)
{
x.resize(pointsNum, 0.0);
w.resize(pointsNum, 0.0);
const int MAXIT=10000;
const MDOUBLE EPS=1.0e-6;
int i,its,j;
MDOUBLE ai,p1,p2,p3,pp,z=0.0,z1;
int n= x.size();
for (i=0;i<n;i++) {
//loops over the desired roots
if (i == 0) { //initial guess for the smallest root
z=(1.0+alf)*(3.0+0.92*alf)/(1.0+2.4*n+1.8*alf);
} else if (i == 1) {//initial guess for the second smallest root
z += (15.0+6.25*alf)/(1.0+0.9*alf+2.5*n);
} else { //initial guess for the other roots
ai=i-1;
z += ((1.0+2.55*ai)/(1.9*ai)+1.26*ai*alf/
(1.0+3.5*ai))*(z-x[i-2])/(1.0+0.3*alf);
}
for (its=0;its<MAXIT;its++) { //refinement by Newton's method
p1=1.0;
p2=0.0;
for (j=0;j<n;j++) { //Loop up the recurrence relation to get the Laguerre polynomial evaluated at z.
p3=p2;
p2=p1;
p1=((2*j+1+alf-z)*p2-(j+alf)*p3)/(j+1);
}
//p1 is now the desired Laguerre polynomial. We next compute pp, its derivative,
//by a standard relation involving also p2, the polynomial of one lower order.
pp=(n*p1-(n+alf)*p2)/z;
z1=z;
z=z1-p1/pp; //Newton's formula
if (fabs(z-z1) <= EPS)
break;
}
if (its >= MAXIT)
errorMsg::reportError("too many iterations in gaulag");
x[i]=z;
w[i] = -exp(gammln(alf+n)-gammln(MDOUBLE(n)))/(pp*n*p2);
}
}
void GLaguer::GetPhylipLaguer(const int categs, MDOUBLE alpha, Vdouble & points, Vdouble & weights)
{
/* calculate rates and probabilities to approximate Gamma distribution
of rates with "categs" categories and shape parameter "alpha" using
rates and weights from Generalized Laguerre quadrature */
points.resize(categs, 0.0);
weights.resize(categs, 0.0);
long i;
raterootarray lgroot; /* roots of GLaguerre polynomials */
double f, x, xi, y;
alpha = alpha - 1.0;
lgroot[1][1] = 1.0+alpha;
for (i = 2; i <= categs; i++)
{
cerr<<lgroot[i][1]<<"\t";
lgr(i, alpha, lgroot); /* get roots for L^(a)_n */
cerr<<lgroot[i][1]<<endl;
}
/* here get weights */
/* Gamma weights are (1+a)(1+a/2) ... (1+a/n)*x_i/((n+1)^2 [L_{n+1}^a(x_i)]^2) */
f = 1;
for (i = 1; i <= categs; i++)
f *= (1.0+alpha/i);
for (i = 1; i <= categs; i++) {
xi = lgroot[categs][i];
y = glaguerre(categs+1, alpha, xi);
x = f*xi/((categs+1)*(categs+1)*y*y);
points[i-1] = xi/(1.0+alpha);
weights[i-1] = x;
}
}
void GLaguer::lgr(long m, double alpha, raterootarray lgroot)
{ /* For use by initgammacat. Get roots of m-th Generalized Laguerre
polynomial, given roots of (m-1)-th, these are to be
stored in lgroot[m][] */
long i;
double upper, lower, x, y;
bool dwn; /* is function declining in this interval? */
if (m == 1) {
lgroot[1][1] = 1.0+alpha;
} else {
dwn = true;
for (i=1; i<=m; i++) {
if (i < m) {
if (i == 1)
lower = 0.0;
else
lower = lgroot[m-1][i-1];
upper = lgroot[m-1][i];
}
else { /* i == m, must search above */
lower = lgroot[m-1][i-1];
x = lgroot[m-1][m-1];
do {
x = 2.0*x;
y = glaguerre(m, alpha,x);
} while ((dwn && (y > 0.0)) || ((!dwn) && (y < 0.0)));
upper = x;
}
while (upper-lower > 0.000000001) {
x = (upper+lower)/2.0;
if (glaguerre(m, alpha, x) > 0.0) {
if (dwn)
lower = x;
else
upper = x;
}
else {
if (dwn)
upper = x;
else
lower = x;
}
}
lgroot[m][i] = (lower+upper)/2.0;
dwn = !dwn; // switch for next one
}
}
} /* lgr */
double GLaguer::glaguerre(long m, double b, double x)
{ /* Generalized Laguerre polynomial computed recursively.
For use by initgammacat */
long i;
double gln, glnm1, glnp1; /* L_n, L_(n-1), L_(n+1) */
if (m == 0)
return 1.0;
else {
if (m == 1)
return 1.0 + b - x;
else {
gln = 1.0+b-x;
glnm1 = 1.0;
for (i=2; i <= m; i++) {
glnp1 = ((2*(i-1)+b+1.0-x)*gln - (i-1+b)*glnm1)/i;
glnm1 = gln;
gln = glnp1;
}
return gln;
}
}
} /* glaguerre */
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